The basic premise of network model is to represent the elements of system as nodes and to use links to capture the pairwise relationships. High-order interactions mean that there are more than two entities involved at once[1]}, which are ubiquitous in various scenes[2]}, [3]}, [4]}. Capturing and characterizing high-order structures in networks is helpful for revealing network evolution mechanisms. Motivated by the significance of the triangular structures in network clustering and the theory of triadic closure in social networks[5]}, we employ this theory via the increasing of structural order. Similar to the definition in algebraic topology, a set of \(k\)  nodes is called a \((k-1)\) -simplex, and the set with all possible connections is called a \(k\) -clique in graph theory. Likewise, a simplicial complex is a finite set of simplices[6]}, [1]}. As shown in Fig. REF , 0-simplices are nodes, 1-simplices are edges, and 2-simplices are triangles. The simplicial complex \(J\)  is composed of two 2-simplices.
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